The t Test for Two The t Test for Two Independent SamplesIndependent Samples

What Does a t Test for What Does a t Test for Independent Samples Mean?Independent Samples Mean?

We will look at difference scores between We will look at difference scores between two samples.two samples.

A research design that uses a separate A research design that uses a separate sample for each treatment condition (or for sample for each treatment condition (or for each population) is called an independent-each population) is called an independent-measures research design or a between-measures research design or a between-subjects designsubjects design

This is in contrast to repeated measures or This is in contrast to repeated measures or within-subjects designswithin-subjects designs

What Do Our Hypotheses Look What Do Our Hypotheses Look Like For These Tests?Like For These Tests?

Null:Null: HH00: : μμ11 = = μμ22 (No difference between the (No difference between the

population means)population means) Same as Same as μμ11 - - μμ22 = 0 = 0

AlternativeAlternative HH11: : μμ11 ≠ μ ≠ μ22 (There is a mean difference) (There is a mean difference)

Same as Same as μμ11 - μ - μ22 ≠ 0 ≠ 0

What is the Formula for Two What is the Formula for Two Sample t – tests?Sample t – tests?

It is actually very similar to the one sample It is actually very similar to the one sample test…test… t = [(Mt = [(M11 – M – M22) – () – (μμ11 - - μμ22)] / s)] / s(M1 – M2)(M1 – M2)

This says that t is equal to the mean observed This says that t is equal to the mean observed difference minus the mean expected difference minus the mean expected difference all divided by the standard errordifference all divided by the standard error

This begs the question…This begs the question… What is the standard error for two samples?What is the standard error for two samples?

What Is the Standard Error for What Is the Standard Error for Two Samples?Two Samples?

We know that MWe know that M1 1 approximates approximates μμ11 with with

some errorsome error Also, MAlso, M22 approximates approximates μμ22 with some error with some error

Therefore we have two sources of errorTherefore we have two sources of error We pool this error with the following We pool this error with the following

formulaformula ss(M1 – M2) (M1 – M2) = = √[(s√[(s11

22/n/n11) + ) + (s(s2222/n/n22)])]

But There Is a Problem…But There Is a Problem…

Does anyone know the problem with this Does anyone know the problem with this standard error?standard error? It only works for nIt only works for n11 = n = n2.2.

When this isn’t the case we need to use When this isn’t the case we need to use pooled estimates of variance, otherwise pooled estimates of variance, otherwise we will have a biased statistic.we will have a biased statistic.

So what we have to do is pool the So what we have to do is pool the variance.variance.

What does this mean?What does this mean?

What Is the Pooled Variance of What Is the Pooled Variance of Two Samples?Two Samples?

To correct for the bias in the sample To correct for the bias in the sample variances, the independent-measures t variances, the independent-measures t statistic will combine the two sample statistic will combine the two sample variances into a single value called the variances into a single value called the pooled variance.pooled variance.

The formula for pooled variance is:The formula for pooled variance is: sspp

22 = (SS = (SS11 + SS + SS22) / (df) / (df11 + df + df22)) This allows us to calculate an estimate of This allows us to calculate an estimate of

the standard errorthe standard error

What Is Our New Estimate of What Is Our New Estimate of Standard Error?Standard Error?

For this we use the pooled variance in For this we use the pooled variance in place of the sample varianceplace of the sample variance

ss(M1 – M2) (M1 – M2) = = √[(s√[(spp22/n/n11) + ) + (s(spp

22/n/n22)])] What does the pooled standard error tell What does the pooled standard error tell

us?us? It is a measure of the standard discrepancy It is a measure of the standard discrepancy

between a sample statistics (Mbetween a sample statistics (M11 – M – M22) and the ) and the corresponding population parameter (corresponding population parameter (μμ11 - - μμ22))

Now all we need are the df.Now all we need are the df.

How Do We Calculate the df?How Do We Calculate the df?

We need to take into account both We need to take into account both samplessamples dfdf11 = n = n11 – 1 – 1

dfdf22 = n = n22 – 1 – 1

Finally, the dfFinally, the dftot tot = df= df1 1 + df+ df22

An ExampleAn Example

Group 1Group 1 {19, 20, 24, 30, 31, 32, 30, 27, 22, 25}{19, 20, 24, 30, 31, 32, 30, 27, 22, 25} nn11 = 10 = 10 MM11 = 26 = 26 SSSS11 = 200 = 200

Group 2Group 2 {23, 22, 15, 16, 18, 12, 16, 19, 14, 25}{23, 22, 15, 16, 18, 12, 16, 19, 14, 25} nn22 = 10 = 10 MM22 = 18 = 18 SSSS22 = 160 = 160

Step 1: State Your HypothesesStep 1: State Your Hypotheses

NullNull HH00: : μμ11 = = μμ22

AlternativeAlternative HH11: : μμ11 ≠ μ ≠ μ22

State your alphaState your alpha αα = .05 = .05

Step 2: Find tStep 2: Find tcritcrit

First find the dfFirst find the df dfdftottot = df = df1 1 + df+ df22 = 9 + 9 = 18 = 9 + 9 = 18

Find the two tailed critical t value for df = Find the two tailed critical t value for df = 18 and 18 and αα = .05 = .05 ttcritcrit = 2.101 = 2.101

Step 3: Sample Data and Test Step 3: Sample Data and Test StatisticsStatistics

nn11 = 10 = 10 MM11 = 26 = 26 SSSS11 = 200 = 200 nn22 = 10 = 10 MM22 = 18 = 18 SSSS22 = 160 = 160 sspp

22 = (SS = (SS11 + SS + SS22) / (df) / (df11 + df + df22) = 20) = 20 ss(M1 – M2) (M1 – M2) = = √[(s√[(spp

22/n/n11) + ) + (s(spp22/n/n22)] = 2)] = 2

ttobsobs = [(M = [(M11 – M – M22) – () – (μμ11 - - μμ22)] / s)] / s(M1 – M2) (M1 – M2) = 4= 4

Step 4: Make a DecisionStep 4: Make a Decision

Is our observed t (tIs our observed t (tobsobs) greater than, or less ) greater than, or less

than the critical value for t (tthan the critical value for t (tcritcrit)) 4 > 2.1014 > 2.101

Therefore we make the decisionTherefore we make the decision t(18) = 4.00, p<.05t(18) = 4.00, p<.05

Effect SizeEffect Size

d = (Md = (M11 – M – M22) / ) / √s√spp22

rr22 = t = t22/(t/(t22 + df) + df) rr22 = PRE = variability explained by = PRE = variability explained by

treatment / total variabilitytreatment / total variability

Confidence IntervalsConfidence Intervals

Point EstimatePoint Estimate Interval EstimateInterval Estimate

Assumptions!Assumptions! There are always assumptions underlying There are always assumptions underlying

statistical tests.statistical tests. We need to make sure to know these We need to make sure to know these

assumptions to make sure we don’t violate assumptions to make sure we don’t violate them and get misleading results.them and get misleading results.

So what are the t-test assumptions?So what are the t-test assumptions?1.1. The observations within each sample must be The observations within each sample must be

independent.independent.2.2. The two populations from which the samples are The two populations from which the samples are

selected must be normal.selected must be normal.3.3. The two populations from which the samples are The two populations from which the samples are

selected must have equal variances.selected must have equal variances.